Matrix Transfer Technique for Anomalous Diffusion Equation Involving Fractional Laplacian

2021 ◽  
Author(s):  
Minling Zheng ◽  
Zhengmeng Jin ◽  
Fawang Liu ◽  
Vo Anh
Keyword(s):  
Diffusion Equation ◽  
Anomalous Diffusion ◽  
Fractional Laplacian

scholarly journals Caputo Fractional Derivative and Quantum-Like Coherence

Entropy ◽  
2021 ◽  
Vol 23 (2) ◽  
pp. 211
Author(s):  
Garland Culbreth ◽  
Mauro Bologna ◽  
Bruce J. West ◽  
Paolo Grigolini
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Anomalous Diffusion ◽  
Caputo Fractional Derivative ◽  
Quantum Coherence ◽  
Time Derivative ◽  
The Other ◽  
Diffusion Equations ◽  
Self Organization ◽  
Ordinary Time

We study two forms of anomalous diffusion, one equivalent to replacing the ordinary time derivative of the standard diffusion equation with the Caputo fractional derivative, and the other equivalent to replacing the time independent diffusion coefficient of the standard diffusion equation with a monotonic time dependence. We discuss the joint use of these prescriptions, with a phenomenological method and a theoretical projection method, leading to two apparently different diffusion equations. We prove that the two diffusion equations are equivalent and design a time series that corresponds to the anomalous diffusion equation proposed. We discuss these results in the framework of the growing interest in fractional derivatives and the emergence of cognition in nature. We conclude that the Caputo fractional derivative is a signature of the connection between cognition and self-organization, a form of cognition emergence different from the other source of anomalous diffusion, which is closely related to quantum coherence. We propose a criterion to detect the action of self-organization even in the presence of significant quantum coherence. We argue that statistical analysis of data using diffusion entropy should help the analysis of physiological processes hosting both forms of deviation from ordinary scaling.

scholarly journals Numerical methods for solving the multi-term time-fractional wave-diffusion equation

2013 ◽  
Vol 16 (1) ◽  
Author(s):  
Fawang Liu ◽  
Mark Meerschaert ◽  
Robert McGough ◽  
Pinghui Zhuang ◽  
Qingxia Liu
Keyword(s):  
Numerical Methods ◽  
Theoretical Analysis ◽  
Diffusion Equation ◽  
Fractional Derivatives ◽  
Fractional Laplacian ◽  
Numerical Results ◽  
Diffusion Equations ◽  
Time Space ◽  
Methods And Techniques ◽  
Wave Diffusion

AbstractIn this paper, the multi-term time-fractional wave-diffusion equations are considered. The multi-term time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], [1,2), [0,2), [0,3), [2,3) and [2,4), respectively. Some computationally effective numerical methods are proposed for simulating the multi-term time-fractional wave-diffusion equations. The numerical results demonstrate the effectiveness of theoretical analysis. These methods and techniques can also be extended to other kinds of the multi-term fractional time-space models with fractional Laplacian.

Anomalous diffusion equation using a new general fractional derivative within the Miller–Ross kernel

2020 ◽  
Vol 34 (27) ◽  
pp. 2050289
Author(s):  
Yiying Feng ◽  
Jiangen Liu
Keyword(s):  
Analytical Solution ◽  
Diffusion Equation ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Anomalous Diffusion ◽  
Fractional Integral ◽  
Liouville Type

In view of the generalization of Miller–Ross kernel in the sense of Riemann–Liouville type, we propose the new definitions of the general fractional integral (GFI) and general fractional derivative (GFD) to discuss the anomalous diffusion equation, which is distinct from those classic calculus operators. The obtained analytical solution of the application described in the graph is effective and accurate making the use of Laplace transform.

scholarly journals Unique continuation principle for the one-dimensional time-fractional diffusion equation

2019 ◽  
Vol 22 (3) ◽  
pp. 644-657 ◽  
Author(s):  
Zhiyuan Li ◽  
Masahiro Yamamoto
Keyword(s):  
Diffusion Equation ◽  
Anomalous Diffusion ◽  
Function Method ◽  
Unique Continuation ◽  
Fractional Diffusion ◽  
Uniqueness Of Solutions ◽  
One Dimensional ◽  
Continuation Principle ◽  
Time Fractional Diffusion Equation ◽  
The One

Abstract This paper deals with the unique continuation of solutions for a one-dimensional anomalous diffusion equation with Caputo derivative of order α ∈ (0, 1). Firstly, the uniqueness of solutions to a lateral Cauchy problem for the anomalous diffusion equation is given via the Theta function method, from which we further verify the unique continuation principle.

Anomalous diffusion and fractional diffusion equation: anisotropic media and external forces

2005 ◽  
Vol 347 (4-6) ◽  
pp. 160-169 ◽  
Author(s):  
M.F. de Andrade ◽  
E.K. Lenzi ◽  
L.R. Evangelista ◽  
R.S. Mendes ◽  
L.C. Malacarne
Keyword(s):  
Diffusion Equation ◽  
Anomalous Diffusion ◽  
Anisotropic Media ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
External Forces

FRACTIONAL DIFFUSION EQUATION, QUANTUM SUBDYNAMICS AND EINSTEIN'S THEORY OF BROWNIAN MOTION

2007 ◽  
Vol 21 (23n24) ◽  
pp. 3993-3999
Author(s):  
SUMIYOSHI ABE
Keyword(s):  
Brownian Motion ◽  
Quantum Theory ◽  
Fractional Calculus ◽  
Diffusion Equation ◽  
Master Equation ◽  
Anomalous Diffusion ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Quantum Master Equation ◽  
Objective System

The fractional diffusion equation for describing the anomalous diffusion phenomenon is derived in the spirit of Einstein's 1905 theory of Brownian motion. It is shown how naturally fractional calculus appears in the theory. Then, Einstein's theory is examined in view of quantum theory. An isolated quantum system composed of the objective system and the environment is considered, and then subdynamics of the objective system is formulated. The resulting quantum master equation is found to be of the Lindblad type.

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